The diagram shows the graph $y = e^{x}$ for $0 \\leq x \\< 4$ - HSC - SSCE Mathematics Extension 2 - Question 5 - 2018 - Paper 1

To find the volume of the solid formed when the region is rotated about the line $y = -1$, we can use the washer method. The outer radius of the solid formed is the distance from the line $y = -1$ to the curve $y = e^{x}$, which is given by:

$R(x) = e^{x} + 1$

The inner radius is the distance from the line $y = -1$ to the line $y = -1$, which is:

$r(x) = 0$

The volume $V$ can be calculated using the integral:

$V = \pi \int_{0}^{4} \left[R(x)^{2} - r(x)^{2}\right] dx = \pi \int_{0}^{4} \left((e^{x}+1)^{2} - 0^{2}\right) dx$

Thus, the correct integral representing the volume of the solid formed is:

$\pi \int_{0}^{4} (e^{x} + 1)^{2} dx$

This matches option A.